Monotone approximation

Oved Shisha

Vol. 15 (1965), No. 2, 667–671

DOI: 10.2140/pjm.1965.15.667

Abstract

How close can one approximate a monotone function by a monotone polynomial of degree ≦ n, or a convex function by a convex polynomial of degree ≦ n? This leads to the following general question. Let k and n be given, and suppose a real fuction f satisfies f^(k)(x) ≧ 0 throughout a closed, finite interval [a,b]. How close can one approximate f on [a,b] by a polynomial of degree ≦ n whose k-th derivative, too, is ≧ 0 there? We give an answer to the question.

Mathematical Subject Classification

Primary: 41.15

Milestones

Received: 17 March 1964

Published: 1 June 1965